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<span class="kw">reconsider </span><font color="Maroon" title="c1">K</font> = <font color="Maroon">F<sub>1</sub></font>() <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon">F<sub>3</sub></font>(), <font color="Maroon" title="c2">J</font> = <font color="Maroon">F<sub>1</sub></font>() <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon">F<sub>2</sub></font>() as   <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a>   <a href="yellow16.html#V1" title="YELLOW16:attr.1">Poset-yielding</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<div><span class="lab"><font color="Green" title="E41">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">K</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">K</font>] implies for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">P<sub>1</sub></font>[ <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c3">X</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Olive" title="b1">i</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:59_1_1"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">K</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">P<sub>1</sub></font>[ <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c3">X</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Olive" title="b1">i</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[ <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c4">i</font>]</span><br/><a NAME="E2:59_1_1"/>
<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:59_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c4">i</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E1:59_1_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E42">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font></span>)</span>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font>] holds <br/><font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font>] implies <font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:59_1_2"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font></span><br/><a NAME="E2:59_1_2"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
<font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E3:59_1_2"/>
<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:59_1_2"/>
<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E1:59_1_2"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E2:59_1_2"><span class="lab"><font color="Green" title="E44">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E43">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> is   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font></span><br/><a NAME="E1:59_1_3"/>
<font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E2:59_1_3"/>
<font color="Maroon" title="c1">K</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font> is   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:59_1"/>
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">K</font></span>)</span>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">K</font>] holds <br/> <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">K</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Olive" title="b1">X</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c2">J</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="yellow16.html#S1" target="_self" title="YELLOW16:sch.1">YELLOW16:sch 1</a>(<a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E4:59_1"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="txt" href="#E3:59_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:59_1"/>
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>,<font color="Maroon">F<sub>3</sub></font>() <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon">F<sub>1</sub></font>()] holds <br/><font color="Maroon">F<sub>3</sub></font>() <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon">F<sub>1</sub></font>() <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon">F<sub>2</sub></font>() <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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